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Journal of Lie Theory 33 (2023), No. 1, 377--432
Copyright Heldermann Verlag 2023

From Euler Elements and 3-Gradings to Non-Compactly Causal Symmetric Spaces

Vincenzo Morinelli
Department of Mathematics, University Tor Vergata, Rome, Italy

Karl-Hermann Neeb
Department Mathematik, Friedrich-Alexander-Universität, Erlangen-Nürnberg, Germany

Gestur Olafsson
Department of Mathematics, Louisiana State University, Baton Rouge, U.S.A.

We discuss the interplay between causal structures of symmetric spaces and geometric aspects of Algebraic Quantum Field Theory (AQFT). The central focus is the set of Euler elements in a Lie algebra, i.e., elements whose adjoint action defines a 3-grading. In the first half of this article we survey the classification of reductive causal symmetric spaces from the perspective of Euler elements. This point of view is motivated by recent applications in AQFT. In the second half we obtain several results that prepare the exploration of the deeper connection between the structure of causal symmetric spaces and AQFT. In particular, we explore the technique of strongly orthogonal roots and corresponding systems of sl2-subalgebras. Furthermore, we exhibit real Matsuki crowns in the adjoint orbits of Euler elements and we describe the group of connected components of the stabilizer group of Euler elements.

Keywords: Euler element, causal symmetric space, cone field, invariant convex cone.

MSC: 22E45, 81R05, 81T05.

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